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Markus Sköldstam

Analysis of the Phase Space, Asymptotic Behavior and Stability for Tippe Top

The tippe top (TT) is a toy that has the form of a truncated sphere equipped with a little peg. When spun fast on the spherical bottom its center of mass rises above its geometrical center and after a few seconds the top is spinning vertically on the peg. For a long while the counter-intuitive behavior of the tippe top has amused and puzzled many distinguished physicists such as Niels Bohr and Wolfgang Pauli and the work of the last 40 years has resulted in a well accepted model of its motion. Nevertheless there are only a few papers (maybe about 20) discussing the TT and even fewer which include a rigorous mathematical analysis of the equations of motion.

We study the behavior of TT trajectories through the concepts of integrals of motion and of invariant manifolds, following the results of Ebenfeld and Scheck. We study the equations of motion of the tippe top in vector form through a sequence of embedded invariant manifolds to unveil the structure of the top's phase space. The last manifold, consisting of the asymptotic motions, has an attractive character and is analyzed completely. We prove that trajectories in this manifold attracts solutions that are in contact with the plane of support at all times. A full analysis of the stability of asymptotic motions, for all admissible choices of model parameters, has been performed with proofs of the stability conditions given by Ebenfeld and Scheck. A remarkable fact is that the asymptotic trajectories are completely determined by the value of one dynamical variable $\eta_3$ and by an integral of motion $\lambda=L_{3}-\alpha L_{\overline{3}}$, the Jelett's integral. The analysis confirms the known physical behavior of the tippe top.

Sidansvarig: karin.johansson@liu.se
Senast uppdaterad: 2019-12-03